A topological property of the common fixed points set of two multivalued operators. (English) Zbl 1169.54022
Summary: Let \((X,d)\) be a complete metric space and absolute retract for metric spaces. We prove that the common fixed points set of two multivalued operators defined on \(X\), which have the selection property and satisfy a contraction type condition, is an absolute retract for metric spaces.
MSC:
| 54H25 | Fixed-point and coincidence theorems (topological aspects) |
| 54C60 | Set-valued maps in general topology |
| 47H04 | Set-valued operators |
| 47H10 | Fixed-point theorems |
Keywords:
multivalued operator; fixed point; common fixed point; absolute retract; selection propertyReferences:
| [1] | Alicu, M.-C.; Mark, O., Some properties of the fixed points set for multifunctions, Studia Univ. Babeş-Bolyai, Mathematica, 25, 4, 77-79 (1980) · Zbl 0467.54029 |
| [2] | Aubin, J.-P.; Cellina, A., Differential Inclusions. Set-Valued Maps and Viability Theory (1984), Springer-Verlag: Springer-Verlag Berlin · Zbl 0538.34007 |
| [3] | Borsuk, K., Theory of Retracts (1967), PWN — Polish Scientific Publishers: PWN — Polish Scientific Publishers Warsaw · Zbl 0153.52905 |
| [4] | Bressan, A.; Cellina, A.; Fryszkowski, A., A class of absolute retracts in spaces of integrable functions, Proc. Amer. Math. Soc., 112, 2, 413-418 (1991) · Zbl 0747.34014 |
| [5] | Górniewicz, L.; Marano, S. A.; Ślosarski, M., Fixed points of contractive multivalued maps, Proc. Amer. Math. Soc., 124, 9, 2675-2683 (1996) · Zbl 0863.47040 |
| [6] | Michael, E., Continuous selections I, Ann. Math., 63, 2, 361-382 (1956) · Zbl 0071.15902 |
| [7] | Petruşel, A., Operatorial Inclusions (2002), House of the Book of Science: House of the Book of Science Cluj-Napoca · Zbl 1057.47004 |
| [8] | Popa, V., A common fixed point theorem for a sequence of multifunctions, Studia Univ. Babeş-Bolyai, Mathematica, 24, 2, 39-41 (1979) · Zbl 0423.54040 |
| [9] | Ricceri, B., Une propriété topologique de l’ensemble des points fixes d’une contraction multivoque à valeurs convexes, Atti Acc. Lincei Rend., 81, 8, 283-286 (1987) · Zbl 0666.47030 |
| [10] | Schirmer, H., Properties of the fixed point set of contractive multi-functions, Canad. Math. Bull., 13, 2, 169-173 (1970) · Zbl 0199.57701 |
| [11] | Sîntămărian, A., Fixed points and common fixed points for some multivalued operators, Fixed Point Theory, 5, 1, 137-145 (2004) · Zbl 1071.47054 |
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